Chudnovsky, MariaSeymour, PaulSpirkl, SophieZhong, Mingxian2022-08-122022-08-122018-08https://doi.org/10.1016/j.disc.2018.04.020http://hdl.handle.net/10012/18513The final publication is available at Elsevier via https://doi.org/10.1016/j.disc.2018.04.020. © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The graphs with no five-vertex induced path are still not understood. But in the triangle-free case, we can do this and one better; we give an explicit construction for all triangle-free graphs with no six-vertex induced path. Here are three examples: the 16-vertex Clebsch graph, the graph obtained from an 8-cycle by making opposite vertices adjacent, and the graph obtained from a complete bipartite graph by subdividing a perfect matching. We show that every connected triangle-free graph with no six-vertex induced path is an induced subgraph of one of these three (modulo some twinning and duplication).enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/induced subgraphinduced pathArticle